Unified representation of the family of L-functions

Abstract

The aim of this paper is to unify the family of L-functions. By using the generating functions of the Bernoulli, Euler and Genocchi polynomials, we construct unification of the L-functions. We also derive new identities related to these functions. We also investigate fundamental properties of these functions.

AMS Subject Classification:11B68, 11S40, 11S80, 26C05, 30B40.

1 Introduction

The theory of the family of L-functions has become a very important part in the analytic number theory. In this paper, using a new type generating function of the family of special numbers and polynomials, we construct unification of the L-functions.

Throughout this presentation, we use the following standard notions $\mathbb{N}=\{1,2,\dots \}$, ${\mathbb{N}}_0=\{0,1,2,\dots \}=\mathbb{N}\cup \{0\}$, ${\mathbb{Z}}^{+}=\{1,2,3,\dots \}$, ${\mathbb{Z}}^{-}=\{-1,-2,\dots \}$. Also, as usual ℤ denotes the set of integers, ℝ denotes the set of real number and ℂ denotes the set of complex numbers. We assume that $ln(z)$ denotes the principal branch of the multi-valued function $ln(z)$ with the imaginary part $\mathrm{\Im }(ln(z))$ constrained by $-\pi <\mathrm{\Im }(ln(z))\le \pi$.

Recently, the first author [1] introduced and investigated the following generating functions which give a unification of the Bernoulli polynomials, Euler polynomials and Genocchi polynomials:

$$g_{a,b}(x;t,k,\beta ):=\frac{2^{1-k}{t}^k{e}^{tx}}{{\beta }^b{e}^t-a^b}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\beta }(x;k,a,b)\frac{t^n}{n!},$$

(1)

where ($|t|<2\pi$ when $\beta =a$; $|t|<|blog(\frac{\beta }{a})|$ when $\beta \ne a$; $k\in {\mathbb{N}}_0$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\{0\}$).

For the special values of a, b, k, b and β, the polynomials ${\mathcal{Y}}_{n,\beta }(x;k,a,b)$ provide us with a generalization and unification of the classical Bernoulli polynomials, Euler polynomials and Genocchi polynomials and also of the Apostol-type (Apostol-Bernoulli, Apostol-Euler, Apostol-Genocchi) polynomials.

Remark 1.1 If we set $k=a=b=1$ in (1), we get a special case of the generalized Bernoulli polynomials ${\mathcal{Y}}_{n,\beta }(x,k,1,1)$, that is, the so-called Apostol-Bernoulli polynomials ${\mathcal{B}}_n(x,\beta )$ generated by

$$\frac{t}{\beta e^t-1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_n(x,\beta )\frac{t^n}{n!}$$

(cf. [1–28]).

Remark 1.2 By substituting $k+1=-a=b=1$ in (1), we are led to Apostol-Euler polynomials ${\mathcal{E}}_n(x,\beta )$ which are defined by means of the following generating function:

$$\frac{2}{\beta e^t+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_n(x,\beta )$$

(cf. [1–28]).

Remark 1.3 Setting $k=-a=b=1$ into (1), we get the Apostol-Genocchi polynomials ${\mathcal{G}}_n(x,\beta )$ which are defined by means of the following generating function:

$$\frac{2t}{\beta e^t+1}{e}^{xt}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{G}}_n(x,\beta )\frac{t^n}{n!}$$

(cf. [1–28]).

In terms of a Dirichlet character χ of conductor $f\in \mathbb{N}$, Ozden et al. [16] extended and investigated the generating functions of the generalized Bernoulli, Euler and Genocchi numbers and the generalized Bernoulli, Euler and Genocchi polynomials with parameters a, b, β and k. Such χ-extended polynomials and χ-extended numbers are useful in many areas of mathematics and mathematical physics.

Definition 1.4 (Ozden et al. [[16], p.2783])

Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi numbers ${\mathcal{Y}}_{n,\chi ,\beta }(k,a,b)$ and the aforementioned χ-extended generalized Bernoulli-Euler-Genocchi polynomials ${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)$ are given by the following generating functions:

$$F_{\chi ,\beta }(t;k,a,b)=2^{1-k}{t}^k\sum _{j=1}^f\frac{\chi (j){(\frac{\beta }{a})}^{bj}{e}^{jt}}{{\beta }^{bf}{e}^{ft}-a^{bf}}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\chi ,\beta }(k,a,b)\frac{t^n}{n!},$$

(2)

where ($|t|<2\pi$ when $\beta =a$; $|t|<|blog(\frac{\beta }{a})|$ when $\beta \ne a$; $k\in {\mathbb{N}}_0$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\{0\}$) and

$${\mathfrak{H}}_{\chi ,\beta }(x,t;k,a,b)=F_{\chi ,\beta }(t,k;a,b)e^{tx}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)\frac{t^n}{n!}$$

(3)

($|t|<2\pi$ when $\beta =a$; $|t|<|blog(\frac{\beta }{a})|$ when $\beta \ne a$; $k\in {\mathbb{N}}_0$; $\beta \in \mathbb{C}$ ($|\beta |<1$); $a,b\in \mathbb{C}\mathrm{\setminus }\{0\}$).

Remark 1.5 Substituting $k=a=b=\beta =1$ into (2), we are led immediately to the generating function of the generalized Bernoulli numbers which are defined by means of the following generating function:

$$\sum _{j=1}^f\frac{\chi (j)t{e}^{jt}}{e^{ft}-1}=\sum _{n=0}^{\mathrm{\infty }}{B}_{n,\chi }\frac{t^n}{n!}$$

(4)

(cf. [1–26]).

2 Unification of the L-functions

Our aim in this section is to apply the Mellin transformation to the generating function (3) of the polynomials ${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)$ in order to construct a unification of the various members of the family of the L-functions and to thereby interpolate ${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)$ for negative integer values of n.

Throughout this section, we assume that $\beta \in \mathbb{C}$ with $|\beta |<1$ and $s\in \mathbb{C}$.

By substituting (1) into (2), we obtain the following functional equation:

$$F_{\chi ,\beta }(t;k,a,b)=\frac{1}{f^k}\sum _{j=1}^f\chi (j){\left(\frac{\beta }{a}\right)}^{bj}{g}_{a^f,b}(\frac{j}{f},tf;k,{\beta }^f).$$

(5)

By using this functional equation, we arrive at the following theorem.

Theorem 2.1 Let χ be a Dirichlet character of conductor f. Then we have

$${\mathcal{Y}}_{n,\chi ,\beta }(k,a,b)=f^{n-k}\sum _{j=1}^f\chi (j){\left(\frac{\beta }{a}\right)}^{bj}{\mathcal{Y}}_{n,{\beta }^f}(\frac{j}{f};k,a^f,b).$$

(6)

By using (5), we modify (3) as follows:

$${\mathfrak{H}}_{\chi ,\beta }(x,t;k,a,b)=\frac{1}{f^k}\sum _{j=1}^f\chi (j){\left(\frac{\beta }{a}\right)}^{bj}{g}_{a^f,b}(\frac{j+x}{f},tf;k,{\beta }^f).$$

(7)

By using (7), we derive the following result.

Corollary 2.2 Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

$${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)=f^{n-k}\sum _{j=1}^f\chi (j){\left(\frac{\beta }{a}\right)}^{bj}{\mathcal{Y}}_{n,{\beta }^f}(\frac{j+x}{f};k,a^f,b).$$

(8)

By applying the Mellin transformation to the generating function (1), Ozden et al. [[16], p.2784 Equation (4.1)] gave an integral representation of the unified zeta function ${\zeta }_{\beta }(s,x;k,a,b)$:

$${\zeta }_{\beta }(s,x;k,a,b)=\frac{1}{\mathrm{\Gamma }(s)}{\int }_0^{\mathrm{\infty }}{t}^{s-k-1}{g}_{a,b}(x;-t;k,\beta )dt(min\{\mathrm{\Re }(s),\mathrm{\Re }(x)\}>0),$$

(9)

where the additional constraint $\mathrm{\Re }(x)>0$ is required for the convergence of the infinite integral, which is given in (9), at its upper terminal. By making use of the above integral representation, Ozden et al. [[16], p.2784 Equation (4.1)] defined the unified zeta function ${\zeta }_{\beta }(s,x;k,a,b)$ as follows:

$${\zeta }_{\beta }(s,x;k,a,b)={(-\frac{1}{2})}^{k-1}\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^{bm}}{a^{b(m+1)}{(m+x)}^s}(\beta \in \mathbb{C}(|\beta |<1);s\in \mathbb{C}(\mathrm{\Re }(s)>1)).$$

(10)

By applying the Mellin transformation to the generating function (7), we have the following integral representation of the unified two-variable L-functions $L_{\chi ,\beta }(s,x;k,a,b)$:

(11)

in terms of the generating function ${\mathfrak{H}}_{\chi ,\beta }(x,t;k;a,b)$ defined in (7). By substituting (9) into ( 11), we obtain

$$L_{\chi ,\beta }(s,x;k,a,b)=\frac{1}{f^{k+s}}\sum _{j=1}^f\chi (j){\left(\frac{\beta }{a}\right)}^{bj}{\zeta }_{{\beta }^f}(s,\frac{j+x}{f};k,a^f,b)$$

(12)

where ($\beta \in \mathbb{C}$ ($|\beta |<1$); $s\in \mathbb{C}$ ($\mathrm{\Re }(s)>1$)).

Consequently, by making use of (10) and (12), we are ready to define a two-variable unification of the Dirichlet-type L-functions $L_{\chi ,\beta }(s,x;k,a,b)$ as follows.

Definition 2.3 Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. For $s,\beta \in \mathbb{C}$ ($|\beta |<1$), we define a two-variable unified L-function $L_{\chi ,\beta }(s,x;k,a,b)$ by

$$L_{\chi ,\beta }(s,x;k,a,b)=f^{-k}{(-\frac{1}{2})}^{k-1}\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^{bm}\chi (m)}{a^{b(m+f)}{(m+x)}^s}(\beta \in \mathbb{C}(|\beta |<1);\mathrm{\Re }(s)>1).$$

(13)

Remark 2.4 If we substitute $x=1$ into (13), we get the unified L-function

$$L_{\chi ,\beta }(s;k,a,b):=L_{\chi ,\beta }(s,1;k,a,b)$$

by

$$L_{\chi ,\beta }(s;k,a,b)=f^{-k}{(-\frac{1}{2})}^{k-1}\sum _{m=1}^{\mathrm{\infty }}\frac{{\beta }^{bm}\chi (m)}{a^{b(m+f)}{m}^s},$$

where ($\mathrm{\Re }(s)>1$, $\beta \in \mathbb{C}$ ($|\beta |<1$)).

Remark 2.5 Upon substituting $k=a=b=1$ and $\beta =\frac{\xi }{u}$ into (13), we arrive at the interpolation function for twisted generalized Eulerian numbers and polynomials, which is given as follows:

$$l_1(\frac{u}{\xi },s,\chi )=L_{\chi ,\frac{\xi }{u}}(s,x;1,1,1),$$

where, for a positive integer r, ξ is the r th root of 1.

$$l_1(\frac{u}{\xi },s;\chi )=\sum _{m=0}^{\mathrm{\infty }}{\left(\frac{\xi }{u}\right)}^m\frac{\chi (m)}{{(m+x)}^s}$$

(cf. [18]).

Remark 2.6 Substituting $x=1$ into (13), we get a unification of the L-functions

$$L_{\chi ,\beta }(s,1;k,a,b)=L_{\chi ,\beta }(s;k,a,b).$$

Substituting $\chi \equiv 1$ into (13), we get a unification ${\zeta }_{\beta }(s,x;k,a,b)$ of the Hurwitz-type zeta function which is given in (10). We also note that both the Hurwitz (or generalized) zeta function

$$\zeta (s,x)={\zeta }_1(s,x;1,1,1)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{{(n+x)}^s}$$

(cf. [27, 28]) and the Riemann zeta function

$$\zeta (s)={\zeta }_1(s,1;1,1,1)=\sum _{n=1}^{\mathrm{\infty }}\frac{1}{n^s}$$

are obvious special cases of the unified zeta function ${\zeta }_{\beta }(s,x;k,a,b)$ (cf. [16, 27, 28]). The relationship between the unified zeta function and the Hurwitz-Lerch zeta function $\mathrm{\Phi }(z,s,a)$ was given by Ozden et al. [16]:

$${\zeta }_{\beta }(s,x;k,a,b):={(-\frac{1}{2})}^{k-1}{a}^{-b}\mathrm{\Phi }(\frac{{\beta }^b}{a^b},s,x),$$

(14)

where the Hurwitz-Lerch zeta function is defined by

$$\mathrm{\Phi }(z,s,x)=\sum _{n=0}^{\mathrm{\infty }}\frac{z^n}{{(n+x)}^s},$$

which converges for ($x\in {\mathbb{C}\mathrm{╲}\mathbb{Z}}_0^{-}$, $s\in \mathbb{C}$ when $|z|<1$; $\mathrm{\Re }(s)>1$ when $|z|=1$), where as usual

$${\mathbb{Z}}_0^{-}={\mathbb{Z}}^{-}\cup \{0\}$$

(cf. [27, 28]).

A relationship between the functions $L_{\chi ,\beta }(s,x;k,a,b)$ and ${\zeta }_{\beta }(s,x;k,a,b)$ is provided by the next theorem.

Theorem 2.7 Let $s\in \mathbb{C}$. Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

$$L_{\chi ,\beta }(s,x;k,a,b)=f^{-s-k}\sum _{j=1}^f{\left(\frac{\beta }{a}\right)}^{jb}\chi (j){\zeta }_{{\beta }^f}(s,\frac{j+x}{f};k,a^f,b).$$

(15)

Proof Substituting $m=nf+j$, $j=1,2,\dots ,f$, $n=0,\dots ,\mathrm{\infty }$ into (13), we obtain

$$L_{\chi ,\beta }(s,x;k,a,b)={(-\frac{1}{2})}^{k-1}{f}^{-s-k}\sum _{j=1}^f{\left(\frac{\beta }{a}\right)}^{jb}\chi (j)\sum _{n=0}^{\mathrm{\infty }}\frac{{\beta }^{bnf}}{a^{bnf}{(n+\frac{j+x}{f})}^s}.$$

After some algebraic manipulations, we arrive at the desired result. □

Remark 2.8 Substituting $a=b=k=1$ into (13), we have

$$L_{\chi ,\beta }(s,x;1,1,1)=\sum _{m=0}^{\mathrm{\infty }}\frac{{\beta }^m\chi (m)}{{(m+x)}^s}(\mathrm{\Re }(s)>1,\beta \in \mathbb{C}(|\beta |<1))$$

which interpolates the Apostol-Bernoulli polynomials attached to the Dirichlet character, which are given by means of the following generating functions:

$$\sum _{j=1}^f\frac{\chi (j)t{\beta }^j{e}^{t(j+x)}}{{\beta }^f{e}^{tf}-1}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{B}}_{n,\chi }(x,\beta )\frac{t^n}{n!}.$$

Let f be an odd integer. If we set $a=-1$ and $k=0$ into (13), then we have

$$L_{\chi ,\beta }(s,x;1,-1,1)=2\sum _{m=1}^{\mathrm{\infty }}{(-1)}^m\frac{\chi (m){\beta }^m}{{(m+x)}^s}(\mathrm{\Re }(s)>1,\beta \in \mathbb{C}(|\beta |<1)),$$

which interpolate the Apostol-Euler polynomials attached to the Dirichlet character, which are defined by the following generating functions:

$$\sum _{j=1}^f\frac{2\chi (j){\beta }^j{e}^{t(j+x)}}{{\beta }^f{e}^{tf}+1}=\sum _{n=0}^{\mathrm{\infty }}{\mathcal{E}}_{n,\chi }(x,\beta )\frac{t^n}{n!}$$

(cf. [1–29]).

By using (15) and (14), we arrive at the following result.

Corollary 2.9 Let $s\in \mathbb{C}$. Let χ be a Dirichlet character of conductor $f\in \mathbb{N}$. Then we have

$$L_{\chi ,\beta }(s,x;k,a,b)={(-\frac{1}{2})}^{k-1}{a}^{-fb}{f}^{-s-k}\sum _{j=1}^f{\left(\frac{\beta }{a}\right)}^{jb}\chi (j)\mathrm{\Phi }(\frac{{\beta }^{fb}}{a^{fb}},s,\frac{j+x}{f}).$$

Theorem 2.10 Let χ be a Dirichlet character of conductor f. Let n be a positive integer. Then we have

$$L_{\chi ,\beta }(1-n,x;k,a,b)=\frac{{(-1)}^k}{f}\frac{(n-1)!}{(n+k-1)!}{\mathcal{Y}}_{n+k-1,\chi ,\beta }(x;k,a,b).$$

(16)

Proof By substituting $s=1-n$ into (15), we get

$$L_{\chi ,\beta }(1-n,x;k,a,b)=f^{n-1-k}\sum _{j=1}^f{\left(\frac{\beta }{a}\right)}^{jb}\chi (j){\zeta }_{{\beta }^f}(1-n,\frac{j+x}{f};k,a^f,b).$$

By using Theorem 7 in [16], we get

By substituting (8) into the above, we arrive at the desired result. □

Remark 2.11 The two-variable Dirichlet L-function and the Dirichlet L-function are obvious special cases of the unified Dirichlet-type L-functions $L_{\chi ,\beta }(s,x;k,a,b)$ defined by (13). We thus have (cf. [13])

$$L(s,x;\chi )=\sum _{m=0}^{\mathrm{\infty }}\frac{\chi (m)}{{(m+x)}^s}$$

and

$$L(s;\chi )=\sum _{m=1}^{\mathrm{\infty }}\frac{\chi (m)}{m^s},$$

where $\mathrm{\Re }(s)>1$. By analytic continuation, this function can be extended to a meromorphic function on the whole complex plane. We have

$$L(1-n;\chi )=-\frac{B_{n,\chi }}{n},$$

where $n\in {\mathbb{Z}}^{+}$ and $B_{n,\chi }$, the usual generalized Bernoulli number, is defined by (4). The Dirichlet L-function is used to prove the theorem on primes in arithmetic progressions. Dirichlet shows that $L(s;\chi )$ is non-zero at $s=1$. Furthermore, if χ is a principal character, then the corresponding Dirichlet L-function has a simple pole at $s=1$ (cf. [6, 7, 9, 18, 24, 27, 28, 30, 31]).

3 Applications

In this section, by using (16) and the following formula, which was proved by Ozden et al. [[16], Theorem 5, Equation (3.10)]

$${\mathcal{Y}}_{n,\chi ,\beta }(x;k,a,b)=\sum _{j=0}^n\left(\genfrac{}{}{0ex}{}{n}{j}\right)x^{n-j}{\mathcal{Y}}_{j,\chi ,\beta }(k,a,b),$$

(17)

we construct a meromorphic function involving a unified family of L-functions. Therefore, using (16) and (17),

$$L_{\chi ,\beta }(1-n,x;k,a,b)=\frac{x^{n+k-1}}{f{\prod }_{l=0}^{k-1}(n+l)}\sum _{j=0}^{n+k-1}\left(\genfrac{}{}{0ex}{}{n+k-1}{j}\right)\frac{1}{x^j}{\mathcal{Y}}_{j+k-1,\chi ,\beta }(k,a,b).$$

From the above equation, we arrive at the following theorem.

Theorem 3.1 Let $x\ne 0$. Let χ be a Dirichlet character of conductor f. Then we have

$$L_{\chi ,\beta }(s,x;k,a,b)=\frac{x^{k-s}}{f{\prod }_{l=0}^{k-1}(s-1-l)}\sum _{j=0}^{\mathrm{\infty }}\left(\genfrac{}{}{0ex}{}{k-s}{j}\right)\frac{1}{x^j}{\mathcal{Y}}_{j+k-1,\chi ,\beta }(k,a,b).$$

The function $L_{\chi ,\beta }(s,x;k,a,b)$ is an analytic function at $s=0$. We now compute the value of this function at this point as follows:

$$L_{\chi ,\beta }(0,x;k,a,b)=\frac{x^k}{{(-1)}^{k}f{\prod }_{l=0}^{k-1}(1+l)}\sum _{j=0}^k\left(\genfrac{}{}{0ex}{}{k}{j}\right)\frac{1}{x^j}{\mathcal{Y}}_{j+k-1,\chi ,\beta }(k,a,b).$$

The function $L_{\chi ,\beta }(s,x;k,a,b)$ is a meromorphic function. This function has simple poles which are

$$s=1,2,3,\dots ,k.$$

The residues of this function at the simple poles at $s=1$ and $s=k$ are given, respectively, as follows:

$${Res}_{s=1}\{L_{\chi ,\beta }(s,x;k,a,b)\}=\frac{x^{k-1}}{f{(-1)}^k{\prod }_{l=0}^{k-1}(2+l)}\sum _{j=0}^{k-1}\left(\genfrac{}{}{0ex}{}{k-1}{j}\right)\frac{1}{x^j}{\mathcal{Y}}_{j+k-1,\chi ,\beta }(k,a,b)$$

and

$${Res}_{s=k}\{L_{\chi ,\beta }(s,x;k,a,b)\}=\frac{{\mathcal{Y}}_{k-1,\chi ,\beta }(k,a,b)}{f{\prod }_{l=0}^{k-2}(k-1-l)}.$$

Remark 3.2 Simsek (cf. [20, 21]) defined a twisted two-variable L-function $L_{\xi ,q}^{(h)}(s,x;\chi )$ as follows:

$$L_{\xi ,q}^{(h)}(s,x;\chi )=\sum _{m=0}^{\mathrm{\infty }}\frac{\chi (m){\varphi }_{\xi }(m)q^{hm}}{{(x+m)}^s}-\frac{log{q}^h}{s-1}\sum _{m=0}^{\mathrm{\infty }}\frac{\chi (m){\varphi }_{\xi }(m)q^{hm}}{{(x+m)}^{s-1}},$$

where $q\in \mathbb{C}$ ($|q|<1$); ${\xi }^r=1$ ($r\in \mathbb{Z}$); $\xi \ne 1$. Observe that if $\xi =1$, then $L_{\xi ,q}^{(h)}(s,x;\chi )$ is reduced to the work of Kim [9].

Relationship between the function $L_{\chi ,\beta }(s,x;k,a,b)$ and $L_{\xi ,q}^{(h)}(s,x;\chi )$ is given as the following result.

Corollary 3.3 Let χ be a Dirichlet character of conductor f. Then we have

$$L_{1,\frac{{\beta }^b}{a^b}}^{(b)}(s,x;\chi )={(-2)}^k{a}^{bf}{f}^k(L_{\chi ,\beta }(s,x;k,a,b)-\frac{log{q}^h}{s-1}{L}_{\chi ,\beta }(s-1,x;k,a,b)).$$

We conclude our present investigation by remarking that the existing literature contains several interesting generalizations and extensions of the Hurwitz-Lerch zeta function $\mathrm{\Phi }(z,s,a)$, Hurwitz zeta function $\zeta (s,x)$ and L-function (cf. [1–30]); see also the references cited in each of these earlier works.